15-85 R. Campoamor-Stursberg

Deformations of Lagrangian systems preserving a fixed subalgebra of Noether symmetries (449K, PDF) Aug 25, 15

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Abstract. Systems of second-order ordinary differential equations admitting a Lagrangian formulation are deformed requiring that the extended Lagrangian preserves a fixed subalgebra of Noether symmetries of the original system. For the case of the simple Lie algebra $rak{sl}(2,\mathbb{R})$, this provides non-linear systems with two independent constants of the motion quadratic in the velocities. In the case of scalar differential equations, it is shown that equations of Pinney-type arise as the most general deformation of the time-dependent harmonic oscillator preserving a $rak{sl}(2,\mathbb{R})$-subalgebra. The procedure is generalized naturally to two dimensions. In particular, it is shown that any deformation of the time-dependent harmonic oscillator in two dimensions that preserves a $rak{sl}(2,\mathbb{R})$ subalgebra of Noether symmetries is equivalent to a generalized Ermakov-Ray-Reid system that satisfies the Helmholtz conditions of the Inverse Problem of Lagrangian Mechanics. Application of the procedure to other types of Lagrangians is illustrated.

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